With the end-of-statement period, I agree you can make that case, but it's done with other punctuation as well, since the American rule is more of a typographical rule than a semantic one (you put low-on-the-line punctuation before high-on-the-line punctuation).

For example:

  British: "I would not", he objected, "enjoy that".
  American: "I would not," he objected, "enjoy that."

The period is arguably part of the quoted statement as you say, but the comma definitely isn't.